Hey guys, let's do an example about AC circuits. Current and voltage in an AC circuit are graphed in the following figure. What are the functions that describe these values? Okay. So just remember that voltage as a function of time in AC circuits is going to be equal to some maximum voltage produced by the source times cosine of omega t where omega is the frequency of the source, and I of t, the current produced by the source, is going to be some i_max, which is the maximum current produced by the source, times cosine of omega t. So in order to find these functions, all we need to do is find what these maximum values are, right? And then what the angular frequency of this oscillation is. Once we know those three values, right, the angular frequency for both of these functions is going to be the same. Once we know those three values we can plug them into our functions and be done with it. Okay?

Now remember that the maximum voltage and the maximum current according to these equations right above me are just the amplitudes of these oscillations. So what's the amplitude of the voltage oscillation? 11 volts. This is V_max. What's the amplitude of the current oscillation? It's 2 and a half amps. This is actually negative i_max. That's why this says negative 2 and a half amps because you're at the negative amplitude. The only question remaining is what's the angular frequency? Well, we are told that from this point up here to this point down here takes 0.05 seconds. Not half a second, 0.05 seconds. Okay?

Well, this distance right here is half of a cycle. A full cycle would be starting from the amplitude, coming down to the negative amplitude, and going back up to the positive amplitude where you started. Going from to the negative amplitude is half of a cycle and that takes 1 half of a period. So that time 0.05 seconds, is actually half of the period. So if you say one half of the period is 0.05 seconds, then we can just multiply this 2 up to the other side, and we can say that the period is 0.1 seconds. Okay. 0.05 times 2 is just 0.1. Now we want to find angular frequency from period. Okay. Remember that the angular frequency is defined as 2πf which is the same as 2π/T. So this is 2π<\/mn>0.1<\/mn> seconds<\/mtext><\/mrow><\/mfrac><\/math> which is gonna be 62.8 inverse seconds. Okay?

So now we know all three of our values. We know that the angular frequency is 62.8 inverse seconds, we know that the maximum voltage is 11 volts, and we know that the maximum current is 2 and a half amps. So all we have to do is plug in those three values to the two equations above me. And we'll say that the voltage as a function of time is: V(t) = 11 volts × cos(62.8⁻¹ seconds × t). The current as a function of time is the maximum current, which is 2 and a half amps, times cosine of, once again, the angular frequency which is 62.8⁻¹ seconds × t). And these are our answers.

All right, guys. Thanks for watching.